Changes for Content of Mathematics Subject Matter Requirements (SMRs)
Between 2009 and 2013 Versions
This document is a little toolkit for identifying changes between the old and new CTC SMRs.
This could be useful for people who have to resubmit their old subject matter waiver program
approvals.
(Pages 2—8) First, there is a Microsoft Word "track changes" version of the new tables. You can
flip between Review/Final Showing Markup and Original Showing Markup if you want to see
the changes. This version was created by Rebecca Parker at CTC and compiled by Carol Fry
Bohlin.
(Pages 9—14) Second, there is a side-by-side table version of the changes (2009/2013 versions)
which was created by Eric Hsu. CTC claims that small changes are very meaningful, so almost
all wording changes are highlighted and bold.
Corrections welcome!
Eric Hsu
erichsu@sfsu.edu
November 5th, 2013
Document History

Nov 5 2013. CFB adds more Parker documentation.

Nov 4 2013. Original version.
1
1. Tracked Changes for Content of Mathematics Subject Matter Requirements (SMRs)
Between 2009 and 2013 Versions
The following is compiled from emails sent from Rebecca Parker (California Commission on
Teacher Credentialing) to Carol Fry Bohlin on 28 October 2013:
All approved Mathematics subject matter programs (programs that waive the CSET: Math) must
review their courses of study to ensure the programs are teaching the new subject matter
requirements (SMRs) essential to effectively teach students the new California Common Core
State Standards (CCSS). Programs must be teaching the new SMRs this Fall (Fall 2013). In
addition, programs must submit a new Alignment Matrix to the Commission on Teacher
Credentialing (CTC) by June 2014. Please do not revise courses to reflect the students’ CCSS.
The CTC convened expert panels to review the CCSS and identify the skills teachers must have
in order to be able to effectively teach students the CCSS. (English SMPPs are making the same
revisions.)
A revised alignment matrix is available on the (CTC) website at www.ctc.ca.gov/educatorprep/STDS-subject-matter.html. Once on the page, scroll down to find Mathematics. The link to
the new Alignment Matrix is just below the word "Mathematics." The handbook is also
available, but is being updated and, although it contains the updated alignment matrix, it doesn’t
provide information to help complete the matrix.
To update the alignment matrix, for each domain and sub-domain on the matrix, identify the
courses, assignments, assessments, etc. through which your students are learning the new SMRs.
Since your program must be teaching the new SMRs this Fall (Fall 2013), you could reference
those courses, assignments, etc. in the matrix. This task doesn’t require a major reorganization or
a resubmission of your program. In fact, please, do not submit copies of syllabi or anything else
as documentation. The completed alignment matrix will be your assurance to the CTC that your
students are being prepared appropriately.
Faculty need to be very careful to notice the verbs and verb changes. The 2009 standards say
“know” a lot. The 2013 standards say “demonstrate”. The revised SMRs put much greater
emphasis on performance because the CCSS requires more mathematical practice; developing
procedural fluency, etc. The following is taken from the revised Mathematics SMPP Handbook,
Attachment to Standard 3 (about required content). The quote is from the K-12 California
Common Core State Standards for Mathematics document at
www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf :
The Common Core State Standards in Mathematics are comprised of two types of standards;
eight practice standards that are identical for each grade level and content standards that differ
at each grade level. Separating the two types of standards emphasizes both the importance of
“knowing” the content and of “being able to do/use” mathematical knowledge. Overlapping
the types of standards elucidates skills such as “habits of mind” and perseverance, which are
critical to mathematics and to staying the course of ones education through difficult and
challenging times. These are worthy goals for students beginning at the earliest levels.
2
The Standards for Mathematical Practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students. These practices rest on
important “processes and proficiencies” with longstanding importance in mathematics
education. The first of these are the NCTM process standards of problem solving, reasoning
and proof, communication, representation, and connections. The second are the strands of
mathematical proficiency specified in the National Research Council’s report Adding It Up:
"adaptive reasoning, strategic competence, conceptual understanding…, procedural fluency…,
and productive disposition (habitual inclination to see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence and one’s own efficacy)."[1]
For guidance or more information, contact Rebecca Parker rparker@ctc.ca.gov. Submit the
completed alignment matrix to Accreditation@ctc.ca.gov .
3
Domains for Mathematics
Domain 1: Algebra
1.1 Algebraic Structures
a. Demonstrate knowledge of why the real and complex numbers are each a field,
and that particular rings are not fields (e.g., integers, polynomial rings, matrix
rings)
b. Apply basic properties of real and complex numbers in constructing
mathematical arguments (e.g., if a < b and c < 0, then ac > bc)
c. Demonstrate knowledge that the rational numbers and real numbers can be
ordered and that the complex numbers cannot be ordered, but that any
polynomial equation with real coefficients can be solved in the complex field
d. Identify and translate between equivalent forms of algebraic expressions and
equations using a variety of techniques (e.g., factoring, applying properties of
operations)
e. Justify the steps in manipulating algebraic expressions and solving algebraic
equations and inequalities
f. Represent situations and solve problems using algebraic equations and
inequalities
1.2 Polynomial Equations and Inequalities
a. Analyze and solve polynomial equations with real coefficients using:
 the Fundamental Theorem of Algebra
 the Rational Root Theorem for polynomials with integer coefficients
 the Conjugate Root Theorem for polynomial equations with real
coefficients
 the Binomial Theorem
b. Prove and use the Factor Theorem and the quadratic formula for real and
complex quadratic polynomials
c. Solve polynomial inequalities
1.3 Functions
a. Analyze general properties of functions (i.e., domain and range, one-to-one,
onto, inverses, composition, and differences between relations and functions)
and apply arithmetic operations on functions
b. Analyze properties of linear functions (e.g., slope, intercepts) using a variety of
representations
c. Demonstrate knowledge of why graphs of linear inequalities are half planes
and be able to apply this fact
d. Analyze properties of polynomial, rational, radical, and absolute value
functions in a variety of ways (e.g., graphing, solving problems)
e. Analyze properties of exponential and logarithmic functions in a variety of
ways (e.g., graphing, solving problems)
f. Model and solve problems using nonlinear functions
1.4 Linear Algebra
a. Understand and apply the geometric interpretation and basic operations of
4
Coursework,
Assignments,
Assessments
Domains for Mathematics
vectors in two and three dimensions, including their scalar multiples
b. Prove the basic properties of vectors (e.g., perpendicular vectors have zero dot
product)
c. Understand and apply the basic properties and operations of matrices and
determinants (e.g., to determine the solvability of linear systems of equations)
d. Analyze the properties of proportional relationships, lines, linear equations,
and their graphs, and the connections between them
e. Model and solve problems using linear equations, pairs of simultaneous linear
equations, and their graphs
Domain 2: Geometry
2.1 Plane Euclidean Geometry
a. Apply the Parallel Postulate and its implications and justify its equivalents
(e.g., the Alternate Interior Angle Theorem, the angle sum of every triangle is
180 degrees)
b. Demonstrate knowledge of complementary, supplementary, and vertical
angles
c. Prove theorems, justify steps, and solve problems involving similarity and
congruence
d. Apply and justify properties of triangles (e.g., the Exterior Angle Theorem,
concurrence theorems, trigonometric ratios, triangle inequality, Law of Sines,
Law of Cosines, the Pythagorean Theorem and its converse)
e. Apply and justify properties of polygons and circles from an advanced
standpoint (e.g., derive the area formulas for regular polygons and circles
from the area of a triangle)
f. Identify and justify the classical constructions (e.g., angle bisector,
perpendicular bisector, replicating shapes, regular polygons with 3, 4, 5, 6,
and 8 sides)
2.2 Coordinate Geometry
a. Use techniques in coordinate geometry to prove geometric theorems
b. Model and solve mathematical and real-world problems by applying
geometric concepts to two-dimensional figures
c. Translate between the geometric description and the equation for a conic
section
d. Translate between rectangular and polar coordinates and apply polar
coordinates and vectors in the plane
2.3 Three-Dimensional Geometry
a. Demonstrate knowledge of the relationships between lines and planes in three
dimensions (e.g., parallel, perpendicular, skew, coplanar lines)
b. Apply and justify properties of three-dimensional objects (e.g., the volume
and surface area formulas for prisms, pyramids, cones, cylinders, spheres)
c. Model and solve mathematical and real-world problems by applying
geometric concepts to three-dimensional figures
2.4 Transformational Geometry
a. Demonstrate knowledge of isometries in two- and three-dimensional space
(e.g., rotation, translation, reflection), including their basic properties in
5
Coursework,
Assignments,
Assessments
relation to congruence
b. Demonstrate knowledge of dilations (e.g., similarity transformations or
change in scale factor), including their basic properties in relation to
similarity, volume, and area
Domain 3: Number and Quantity
3.1 The Real and Complex Number Systems
a. Demonstrate knowledge of the properties of the real number system and of its subsets
b. Perform operations and recognize equivalent expressions using various representations of real
numbers (e.g., fractions, decimals, exponents)
c. Solve real-world and mathematical problems using numerical and algebraic expressions and
equations
d. Apply proportional relationships to model and solve real-world and mathematical problems
e. Reason quantitatively and use units to solve problems (i.e., dimensional analysis)
f. Perform operations on complex numbers and represent complex numbers and their operations on
the complex plane
3.2 Number Theory
a. Prove and use basic properties of natural numbers (e.g., properties of
divisibility)
b. Use the principle of mathematical induction to prove results in number theory
c. Apply the Euclidean Algorithm
d. Apply the Fundamental Theorem of Arithmetic (e.g., find the greatest common
factor and the least common multiple; show that every fraction is equivalent to a
unique fraction where the numerator and denominator are relatively prime;
prove that the square root of any number, not a perfect square number, is
irrational)
Domain 4: Probability and Statistics
4.1 Probability
a. Prove and apply basic principles of permutations and combinations
b. Illustrate finite probability using a variety of examples and models (e.g., the
fundamental counting principles, sample space)
c. Use and explain the concepts of conditional probability and independence
d. Compute and interpret the probability of an outcome, including the probabilities
of compound events in a uniform probability model
e. Use normal, binomial, and exponential distributions to solve and interpret
probability problems
f. Calculate expected values and use them to solve problems and evaluate outcomes
of decisions
4.2 Statistics
a. Compute and interpret the mean and median of both discrete and continuous
distributions
b. Compute and interpret quartiles, range, interquartile range, and standard deviation
of both discrete and continuous distributions
6
c. Select and evaluate sampling methods appropriate to a task (e.g., random,
systematic, cluster, convenience sampling) and display the results
d. Apply the method of least squares to linear regression
e. Apply the chi-square test
f. Interpret scatter plots for bivariate data to investigate patterns of association
between two quantities (e.g., correlation), including the use of linear models
g. Interpret data on a single count or measurement variable presented in a variety of
formats (e.g., dot plots, histograms, box plots)
h. Demonstrate knowledge of P-values and hypothesis testing
i. Demonstrate knowledge of confidence intervals
Domain 5:
Calculus
5.1 Trigonometry
a. Prove that the Pythagorean Theorem is equivalent to the trigonometric identity
sin2x + cos2x = 1 and that this identity leads to 1 + tan2x = sec2x and 1 + cot2x =
csc2x
b. Prove and apply the sine, cosine, and tangent sum formulas for all real values
c. Analyze properties of trigonometric functions in a variety of ways (e.g., graphing
and solving problems, using the unit circle)
d. Apply the definitions and properties of inverse trigonometric functions (i.e.,
arcsin, arccos, and arctan)
e. Apply polar representations of complex numbers (e.g., DeMoivre's Theorem)
f. Model periodic phenomena with periodic functions
g. Recognize equivalent identities, including applications of the half-angle and
double-angle formulas for sines and cosines
5.2 Limits and Continuity
a. Derive basic properties of limits and continuity, including the Sum, Difference,
Product, Constant Multiple, and Quotient Rules, using the formal definition of a
limit
b. Show that a polynomial function is continuous at a point
c. Apply the intermediate value theorem, using the geometric implications of
continuity
5.3 Derivatives and Applications
a. Derive the rules of differentiation for polynomial, trigonometric, and logarithmic
functions using the formal definition of derivative
b. Interpret the concept of derivative geometrically, numerically, and analytically
(i.e., slope of the tangent, limit of difference quotients, extrema, Newton's
method, and instantaneous rate of change)
c. Interpret both continuous and differentiable functions geometrically and
analytically and apply Rolle's theorem, the mean value theorem, and L'Hôpital's
rule
d. Use the derivative to solve rectilinear motion, related rate, and optimization
problems
e. Use the derivative to analyze functions and planar curves (e.g., maxima, minima,
inflection points, concavity)
f. Solve separable first-order differential equations and apply them to growth and
decay problems
5.4 Integrals and Applications
7
a. Derive definite integrals of standard algebraic functions using the formal
definition of integral
b. Interpret the concept of a definite integral geometrically, numerically, and
analytically (e.g., limit of Riemann sums)
c. Prove the fundamental theorem of calculus, and use it to interpret definite
integrals as antiderivatives
d. Apply the concept of integrals to compute the length of curves and the areas and
volumes of geometric figures
5.5 Sequences and Series
a. Derive and apply the formulas for the sums of finite arithmetic series and finite
and infinite geometric series (e.g., express repeating decimals as a rational
number)
b. Determine convergence of a given sequence or series using standard techniques
(e.g., ratio, comparison, integral tests)
c. Calculate Taylor series and Taylor polynomials of basic functions
8
2. Side-by-Side Changes for Content of Mathematics Subject Matter Requirements
(SMRs) Between 2009 and 2013 Versions
Eric Hsu erichsu@sfsu.edu, November 4th, 2013
NEW Common Core SMRs
OLD SMRs
Domain 1: Algebra
Domain 1. Algebra
1.1 Algebraic Structures
a. Demonstrate knowledge of why the real and
complex numbers are each a field, and that
particular rings are not fields (e.g., integers,
polynomial rings, matrix rings)
b. Apply basic properties of real and complex
numbers in constructing mathematical arguments
(e.g., if a < b and c < 0, then ac > bc)
c. Demonstrate knowledge that the rational
numbers and real numbers can be ordered and
that the complex numbers cannot be ordered,
but that any polynomial equation with real
coefficients can be solved in the complex field
d. Identify and translate between equivalent
forms of algebraic expressions and equations
using a variety of techniques (e.g., factoring,
applying properties of operations)
e. Justify the steps in manipulating algebraic
expressions and solving algebraic equations and
inequalities
1.1 Algebraic Structures
a. Know why the real and complex numbers are each a field,
and that particular rings are not fields (e.g., integers,
polynomial rings, matrix rings)
b. Apply basic properties of real and complex numbers in
constructing mathematical arguments (e.g., if a < b and c < 0,
then ac > bc)
c. Know that the rational numbers and real numbers can be
ordered and that the complex numbers cannot be ordered,
but that any polynomial equation with real coefficients can
be solved in the complex field
f. Represent situations and solve problems using
algebraic equations and inequalities
1.2 Polynomial Equations and Inequalities
1.2 Polynomial Equations and Inequalities
a. Analyze and solve polynomial equations with
real coefficients using:
• the Rational Root Theorem for polynomials with
integer coefficients
b. Prove and use the following:
• the Rational Root Theorem for polynomials with integer
coefficients
• the Fundamental Theorem of Algebra
• the Conjugate Root Theorem for polynomial
equations with real coefficients
(1.2.c). Analyze and solve polynomial equations with real
coefficients using the Fundamental Theorem of Algebra
• the Conjugate Roots Theorem for polynomial equations
with real coefficients
• the Binomial Theorem
b. Prove and use the Factor Theorem and the
quadratic formula for real and complex quadratic
polynomials
• the Binomial Theorem
• the Factor Theorem, • the Quadratic Formula for real and
complex quadratic polynomials
c. Solve polynomial inequalities
1.3 Functions
a. Analyze general properties of functions (i.e.,
domain and range, one-to-one, onto, inverses,
composition, and differences between relations
and functions) and apply arithmetic operations on
functions
1.3 Functions
a. Analyze and prove general properties of functions (i.e.,
domain and range, one-to-one, onto, inverses, composition,
and differences between relations and functions)
9
b. Analyze properties of linear functions (e.g.,
slope, intercepts) using a variety of
representations
c. Demonstrate knowledge of why graphs of
linear inequalities are half planes and be able to
apply this fact
d. Analyze properties of polynomial, rational,
radical, and absolute value functions in a variety
of ways (e.g., graphing, solving problems)
e. Analyze properties of exponential and
logarithmic functions in a variety of ways (e.g.,
graphing, solving problems)
f. Model and solve problems using nonlinear
functions
(1.2.a) Know why graphs of linear inequalities are half planes
and be able to apply this fact (e.g., linear programming)
b. Analyze properties of polynomial, rational, radical, and
absolute value functions in a variety of ways (e.g., graphing,
solving problems)
c. Analyze properties of exponential and logarithmic
functions in a variety of ways (e.g., graphing, solving
problems)
1.4 Linear Algebra
a. Understand and apply the geometric
interpretation and basic operations of vectors in
two and three dimensions, including their scalar
multiples
1.4 Linear Algebra
a. Understand and apply the geometric interpretation and
basic operations of vectors in two and three dimensions,
including their scalar multiples and scalar (dot) and cross
products
b. Prove the basic properties of vectors (e.g.,
perpendicular vectors have zero dot product)
c. Understand and apply the basic properties and
operations of matrices and determinants (e.g., to
determine the solvability of linear systems of
equations)
d. Analyze the properties of proportional
relationships, lines, linear equations, and their
graphs, and the connections between them
e. Model and solve problems using linear
equations, pairs of simultaneous linear equations,
and their graphs
b. Prove the basic properties of vectors (e.g., perpendicular
vectors have zero dot product)
Domain 2: Geometry
Domain 2. Geometry
2.1 Plane Euclidean Geometry
a. Apply the Parallel Postulate and its implications
and justify its equivalents (e.g., the Alternate
Interior Angle Theorem, the angle sum of every
triangle is 180 degrees)
2.1 Parallelism
b. Demonstrate knowledge of complementary,
supplementary, and vertical angles
b. Know that variants of the Parallel Postulate produce nonEuclidean geometries (e.g., spherical, hyperbolic)
c. Understand and apply the basic properties and operations
of matrices and determinants (e.g., to determine the
solvability of linear systems of equations)
a. Know the Parallel Postulate and its implications, and justify
its equivalents (e.g., the Alternate Interior Angle Theorem,
the angle sum of every triangle is 180 degrees)
2.2 Plane Euclidean Geometry
c. Prove theorems, justify steps, and solve
problems involving similarity and congruence
d. Apply and justify properties of triangles (e.g.,
the Exterior Angle Theorem, concurrence
theorems, trigonometric ratios, triangle
inequality, Law of Sines, Law of Cosines, the
Pythagorean Theorem and its converse)
e. Apply and justify properties of polygons and
circles from an advanced standpoint (e.g., derive
the area formulas for regular polygons and circles
from the area of a triangle)
a. Prove theorems and solve problems involving similarity
and congruence
b. Understand, apply, and justify properties of triangles (e.g.,
the Exterior Angle Theorem, concurrence theorems,
trigonometric ratios, Triangle Inequality, Law of Sines, Law of
Cosines, the Pythagorean Theorem and its converse)
c. Understand, apply, and justify properties of polygons and
circles from an advanced standpoint (e.g., derive the area
formulas for regular polygons and circles from the area of a
triangle)
10
f. Identify and justify the classical constructions
(e.g., angle bisector, perpendicular bisector,
replicating shapes, regular polygons with 3, 4, 5,
6, and 8 sides)
2.2 Coordinate Geometry
a. Use techniques in coordinate geometry to
prove geometric theorems
b. Model and solve mathematical and real-world
problems by applying geometric concepts to twodimensional figures
d. Justify and perform the classical constructions (e.g., angle
bisector, perpendicular bisector, replicating shapes, regular
n-gons for n equal to 3, 4, 5, 6, and 8)
e. Use techniques in coordinate geometry to prove geometric
theorems
c. Translate between the geometric description
and the equation for a conic section
d. Translate between rectangular and polar
coordinates and apply polar coordinates and
vectors in the plane
2.3 Three-Dimensional Geometry
a. Demonstrate knowledge of the relationships
between lines and planes in three dimensions
(e.g., parallel, perpendicular, skew, coplanar
lines)
b. Apply and justify properties of threedimensional objects (e.g., the volume and surface
area formulas for prisms, pyramids, cones,
cylinders, spheres)
c. Model and solve mathematical and real-world
problems by applying geometric concepts to
three-dimensional figures
2.3 Three-Dimensional Geometry
2.4 Transformational Geometry
a. Demonstrate knowledge of isometries in twoand three-dimensional space (e.g., rotation,
translation, reflection), including their basic
properties in relation to congruence
b. Demonstrate knowledge of dilations (e.g.,
similarity transformations or change in scale
factor), including their basic properties in relation
to similarity, volume, and area
2.4 Transformational Geometry
Domain 3: Number and Quantity
Domain 3. Number Theory
a. Demonstrate an understanding of parallelism and
perpendicularity of lines and planes in three dimensions
b. Understand, apply, and justify properties of threedimensional objects from an advanced standpoint (e.g.,
derive the volume and surface area formulas for prisms,
pyramids, cones, cylinders, and spheres)
a. Demonstrate an understanding of the basic properties of
isometries in two- and three-dimensional space (e.g.,
rotation, translation, reflection)
b. Understand and prove the basic properties of dilations
(e.g., similarity transformations or change of scale)
3.1 The Real and Complex Number Systems
a. Demonstrate knowledge of the properties of
the real number system and of its subsets
b. Perform operations and recognize equivalent
expressions using various representations of real
numbers (e.g., fractions, decimals, exponents)
c. Solve real-world and mathematical problems
using numerical and algebraic expressions and
equations
d. Apply proportional relationships to model and
solve real-world and mathematical problems
e. Reason quantitatively and use units to solve
problems (i.e., dimensional analysis)
11
f. Perform operations on complex numbers and
represent complex numbers and their operations
on the complex plane
3.2 Number Theory
3.1 Natural Numbers
a. Prove and use basic properties of natural
numbers (e.g., properties of divisibility)
a. Prove and use basic properties of natural numbers (e.g.,
properties of divisibility)
b. Use the principle of mathematical induction to
prove results in number theory
b. Use the Principle of Mathematical Induction to prove
results in number theory
c. Apply the Euclidean Algorithm
d. Apply the Fundamental Theorem of Arithmetic
(e.g., find the greatest common factor and the
least common multiple; show that every fraction
is equivalent to a unique fraction where the
numerator and denominator are relatively prime;
prove that the square root of any number, not a
perfect square number, is irrational)
c. Know and apply the Euclidean Algorithm
Domain 4: Probability and Statistics
Domain 4. Probability and Statistics
4.1 Probability
a. Prove and apply basic principles of
permutations and combinations
b. Illustrate finite probability using a variety of
examples and models (e.g., the fundamental
counting principles, sample space)
c. Use and explain the concepts of conditional
probability and independence
d. Compute and interpret the probability of an
outcome, including the probabilities of
compound events in a uniform probability model
e. Use normal, binomial, and exponential
distributions to solve and interpret probability
problems
f. Calculate expected values and use them to
solve problems and evaluate outcomes of
decisions
4.1 Probability
a. Prove and apply basic principles of permutations and
combinations
4.2 Statistics
4.2 Statistics
a. Compute and interpret the mean and median
of both discrete and continuous distributions
b. Compute and interpret quartiles, range,
interquartile range, and standard deviation of
both discrete and continuous distributions
c. Select and evaluate sampling methods
appropriate to a task (e.g., random, systematic,
cluster, convenience sampling) and display the
results
a. Compute and interpret the mean, median, and mode of
both discrete and continuous distributions
b. Compute and interpret quartiles, range, variance, and
standard deviation of both discrete and continuous
distributions
d. Apply the method of least squares to linear
regression
d. Know the method of least squares and apply it to linear
regression and correlation
e. Apply the chi-square test
f. Interpret scatter plots for bivariate data to
investigate patterns of association between two
quantities (e.g., correlation), including the use of
e. Know and apply the chi-square test
d. Apply the Fundamental Theorem of Arithmetic (e.g., find
the greatest common factor and the least common multiple,
show that every fraction is equivalent to a unique fraction
where the numerator and denominator are relatively prime,
prove that the square root of any number, not a perfect
square number, is irrational)
b. Illustrate finite probability using a variety of examples and
models (e.g., the fundamental counting principles)
c. Use and explain the concept of conditional probability
d. Interpret the probability of an outcome
e. Use normal, binomial, and exponential distributions to
solve and interpret probability problems
c. Select and evaluate sampling methods appropriate to a
task (e.g., random, systematic, cluster, convenience
sampling) and display the results
12
linear models
g. Interpret data on a single count or
measurement variable presented in a variety of
formats (e.g., dot plots, histograms, box plots)
h. Demonstrate knowledge of P-values and
hypothesis testing
i. Demonstrate knowledge of confidence intervals
Domain 5: Calculus
Domain 5. Calculus*
5.1 Trigonometry
a. Prove that the Pythagorean Theorem is
equivalent to the trigonometric identity sin^2 x +
cos^2 x = 1 and that this identity leads to 1 +
tan^2 x = sec^2 x and 1 + cot^2 x = csc^2 x
b. Prove and apply the sine, cosine, and tangent
sum formulas for all real values
c. Analyze properties of trigonometric functions
in a variety of ways (e.g., graphing and solving
problems, using the unit circle)
d. Apply the definitions and properties of inverse
trigonometric functions (i.e., arcsin, arccos, and
arctan)
5.1 Trigonometry
a. Prove that the Pythagorean Theorem is equivalent to the
trigonometric identity sin^2 x + cos^2 x = 1 and that this
identity leads to 1 + tan^2 x = sec^2 x and 1 + cot^2 x = csc^2
x
b. Prove the sine, cosine, and tangent sum formulas for all
real values…
e. Apply polar representations of complex
numbers (e.g., DeMoivre's Theorem)
f. Model periodic phenomena with periodic
functions
g. Recognize equivalent identities, including
applications of the half-angle and double-angle
formulas for sines and cosines
e. Understand and apply polar representations of complex
numbers (e.g., DeMoivre's Theorem)
5.2 Limits and Continuity
a. Derive basic properties of limits and continuity,
including the Sum, Difference, Product, Constant
Multiple, and Quotient Rules, using the formal
definition of a limit
b. Show that a polynomial function is continuous
at a point
5.2 Limits and Continuity
c. Apply the intermediate value theorem, using
the geometric implications of continuity
c. Know and apply the Intermediate Value Theorem, using
the geometric implications of continuity
5.3 Derivatives and Applications
a. Derive the rules of differentiation for
polynomial, trigonometric, and logarithmic
functions using the formal definition of derivative
b. Interpret the concept of derivative
geometrically, numerically, and analytically (i.e.,
slope of the tangent, limit of difference quotients,
extrema, Newton's method, and instantaneous
rate of change)
5.3 Derivatives and Applications
a. Derive the rules of differentiation for polynomial,
trigonometric, and logarithmic functions using the formal
definition of derivative
c. Analyze properties of trigonometric functions in a variety
of ways (e.g., graphing and solving problems)
d. Know and apply the definitions and properties of inverse
trigonometric functions (i.e., arcsin, arccos, and arctan)
(5.1.b). ...and derive special applications of the sum formulas
(e.g., double angle, half angle)
a. Derive basic properties of limits and continuity, including
the Sum, Difference, Product, Constant Multiple, and
Quotient Rules, using the formal definition of a limit
b. Show that a polynomial function is continuous at a point
b. Interpret the concept of derivative geometrically,
numerically, and analytically (i.e., slope of the tangent, limit
of difference quotients, extrema, Newton’s method, and
instantaneous rate of change)
13
c. Interpret both continuous and differentiable
functions geometrically and analytically and apply
Rolle's theorem, the mean value theorem, and
L'Hôpital's rule
c. Interpret both continuous and differentiable functions
geometrically and analytically and apply Rolle’s Theorem, the
Mean Value Theorem, and L’Hopital’s rule
d. Use the derivative to solve rectilinear motion,
related rate, and optimization problems
e. Use the derivative to analyze functions and
planar curves (e.g., maxima, minima, inflection
points, concavity)
f. Solve separable first-order differential
equations and apply them to growth and decay
problems
d. Use the derivative to solve rectilinear motion, related rate,
and optimization problems
5.4 Integrals and Applications
5.4 Integrals and Applications
a. Derive definite integrals of standard algebraic
functions using the formal definition of integral
b. Interpret the concept of a definite integral
geometrically, numerically, and analytically (e.g.,
limit of Riemann sums)
c. Prove the fundamental theorem of calculus,
and use it to interpret definite integrals as
antiderivatives
d. Apply the concept of integrals to compute the
length of curves and the areas and volumes of
geometric figures
a. Derive definite integrals of standard algebraic functions
using the formal definition of integral
5.5 Sequences and Series
a. Derive and apply the formulas for the sums of
finite arithmetic series and finite and infinite
geometric series (e.g., express repeating decimals
as a rational number)
b. Determine convergence of a given sequence or
series using standard techniques (e.g., ratio,
comparison, integral tests)
c. Calculate Taylor series and Taylor polynomials
of basic functions
5.5 Sequences and Series
e. Use the derivative to analyze functions and planar curves
(e.g., maxima, minima, inflection points, concavity)
f. Solve separable first-order differential equations and apply
them to growth and decay problems
b. Interpret the concept of a definite integral geometrically,
numerically, and analytically (e.g., limit of Riemann sums)
c. Prove the Fundamental Theorem of Calculus, and use it to
interpret definite integrals as antiderivatives
d. Apply the concept of integrals to compute the length of
curves and the areas and volumes of geometric figures
a. Derive and apply the formulas for the sums of finite
arithmetic series and finite and infinite geometric series (e.g.,
express repeating decimals as a rational number)
b. Determine convergence of a given sequence or series using
standard techniques (e.g., Ratio, Comparison, Integral Tests)
c. Calculate Taylor series and Taylor polynomials of basic
functions
Domain 6. History of Mathematics*
6.1 Chronological and Topical Development of Mathematics
a. Demonstrate understanding of the development of
mathematics, its cultural connections, and its contributions
to society
b. Demonstrate understanding of the historical development
of mathematics, including the contributions of diverse
populations as determined by race, ethnicity, culture,
geography, and gender
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